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A Brief Introduction to Marginal Analysis for the Micro-Economics

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					JOURNAL OF ECONOMICS AND FINANCE EDUCATION • Volume 7 • Number 2 • Winter 2008                                                      1


A Brief Introduction to Marginal Analysis for the
Micro-Economics Principles Course
Mark L. Burkey1

                                                          ABSTRACT

                     This brief note provides a simple, yet powerful example of how the
                     marginal cost/marginal benefit principle can be used in everyday life.
                     Using the decision of the optimal choice of speed on the highway, this
                     note was developed for use as one of the first readings in an
                     introductory microeconomics course. It is clear in this demonstration
                     that marginal cost is increasing, while marginal benefit is decreasing,
                     and how the intersection of these two curves shows the optimal choice.
                     In addition, shifts in the curves can easily be demonstrated as an
                     introduction to supply and demand.

                                                        Introduction

    While each individual must make their own decision about how fast to drive, some general economic
principles should be kept in mind. Here we describe some factors that enter into the decision.
    In general, we can boil down the costs and benefits of driving faster than the speed limit this way:
Driving faster benefits us by saving time, but results in an increased risk of receiving a ticket and the
associated pecuniary and non-pecuniary costs of a ticket. Additionally, as one drives faster there is an
increased risk of an accident, and increased fuel costs. Most drivers ignore the increased fuel costs: While
the cost of gas is significant, the marginal cost of fuel from driving faster is fairly small.2 When a cost is so
small that we can ignore it for decision-making purposes, economists usually call it a “second-order” cost.
    Let us construct a simple model for driving. Suppose that we need to drive exactly 60 miles, and get
exactly 30 miles to the gallon. The road that we will be driving on has a speed limit of 55. The total cost of
driving this distance will be 2 gallons of gas, plus some wear and tear on the vehicle, plus the value of our
time. Therefore, the benefits of driving faster are reductions in this driving time.

                                  The Marginal Benefit of Driving Faster

   If we drive 55miles per hour, we can calculate the time that the trip will take in the following manner:

   (1) distance = speed*time , therefore
                 distance
   (2) time =               , so time =60/55 = 1.091 hours.
                  speed

It may be more convenient to calculate the time in minutes. To do this, divide the speed by 60:




   1
    Mark L. Burkey, Associate Professor of Economics, Department of Economics and Finance, North Carolina A&T State
   University, Greensboro, NC 27411.
   2
     The wind resistance on a vehicle is proportional to the square of speed, so you will use a little more gas per mile as you drive
   faster. In 1973 Richard Nixon signed a bill into law requiring a national speed limit of 55mph in order to save gasoline. Although
   repealed partially in 1987 and altogether in the 1990’s, in July 2008 a bill was filed in the U.S. Congress to reinstate a national
   speed limit.
JOURNAL OF ECONOMICS AND FINANCE EDUCATION • Volume 7 • Number 2 • Winter 2008                               2


                  60          3600
   (3) time =             =          = 65.45 minutes.
                55 / 60        55

    Should we drive faster? We first need to calculate the marginal benefit of driving faster. That is, we
need to figure out how much additional time we save each time we go a little faster. We can start by
calculating the time difference of increasing/decreasing our speed from 55 miles per hour in increments of
2 miles per hour (using our cruise control):

                          Table 1. Calculating the Marginal Benefit of Driving Faster
                                                     Total time saved
                                                                            Marginal
                           Speed        Time         (compared to 55
                                                                             Benefit
                                                           mph)
                            53        67.92453             -2.47              -2.47
                            55        65.45455            -------            -------
                            57        63.15789              2.30              2.30
                            59        61.01695              4.44              2.14
                            61        59.01639              6.44              2.00
                            63        57.14286              8.31              1.87
                            65        55.38462            10.07               1.76
                            67        53.73134            11.72               1.65
                            69        52.17391            13.28               1.56

    Notice that the marginal benefits are going down as you go faster and faster! This is related to the idea
that you must increase your speed by 100% in order to cut your driving time by only 50%. If we assume
some dollar value for our time, say $.50 per minute, we can calculate the dollar value of the benefit of
increasing our speed.

                                     The Marginal Costs of Driving Faster

    Of course, as you drive faster, your chances of being in a wreck, and the wreck being more serious
increases with speed. Kloeden et al. (1997) found that for each additional 5km/hr (≈ 3 miles per hour)
increase in speed, the probability of being involved in a serious crash involving an injury doubles.
However, the main cost on a driver’s mind is usually the cost of getting a ticket. In some states if you have
no speeding tickets on your record, and you get one speeding ticket for no more than 9 miles per hour over
the speed limit, you must pay the ticket and court costs, but your insurance rates will not go up. However, if
you get a second ticket within 5 years, both tickets will increase your insurance. However, in some states
you can remove one ticket from your record by going to “driving school”.
    So, if a driver has no tickets on their record, they could drive up to 64 miles per hour, and if ticketed,
pay around $150 plus the lost time of being pulled over and mailing in the ticket. Driving a little faster than
64 miles per hour would result in the same penalty, because you can normally go to court and “plea down”
a ticket of up to 14 miles per hour over the speed limit down to 9 miles over. But, the cost increases when
you factor in taking a day off of work and going to court (or hiring an attorney to do this for you).
    However, it is reasonable to expect that the probability of getting a speeding ticket is (almost) zero if
you are within 5 miles of the speed limit. So, increasing your speed from 55 to 60 miles per hour would
generate no additional risk of a ticket. In fact, in many cities, one can go at least 7 miles per hour over the
speed limit, and face almost no chance of getting a ticket. Going 9 miles per hour over greatly increases
your chances of a ticket. It is common practice that police officers will pull you over with certainty (if they
see you) at 11+ miles over the speed limit. Of course, you could drive a long distance without even seeing a
police car. If we were to put the previously mentioned costs and benefits on a graph, they might look like
Figure 1 for a typical driver:
JOURNAL OF ECONOMICS AND FINANCE EDUCATION • Volume 7 • Number 2 • Winter 2008                               3



   Figure 1. Graphing the Marginal Cost and Benefit for Various Speeds


                                                                      Marginal Cost of
        Value                                                         going faster
       of time,
       money,
       risk to
       health


                                                                      Marginal Benefit
                                                                      of going faster



                      55 57 59 61 63 65 67 69 71 73
                                        Speed


    Starting at a speed of 55 miles per hour, we should continue to go faster as long as the additional costs
do not outweigh the additional benefits. According to the graph above, how fast should this driver go?
(The marginal cost and marginal benefit curves intersect at around 65 miles per hour, suggesting that this is
the optimal speed.) However, suppose that the driver depicted in Figure 1 looked down at the speedometer
and saw that he was going 69 miles per hour. What should he do, and why? (He should slow down, because
the additional cost of speeding up to 69 far outweighs the additional benefits.)

                                 Shifts in the Marginal Benefit Curve
    How would your decision change if you were in a big hurry to get somewhere? Perhaps you are late for
work. The value of your time might increase from $.50 to $1.00. Looking at the graph above, what will
happen to the marginal benefit curve? How will this affect your speed? How will this affect your risk of a
ticket? (Your marginal benefit of driving faster would increase, shifting the curve upward. The resulting
optimal speed is higher. You are willing to take a higher risk of a ticket because of the situation.)

                                   Shifts in the Marginal Cost Curve
    Suppose that you will lose your insurance if you get another speeding ticket. How will that affect your
marginal cost curve? (The marginal cost of increasing your speed will be higher, shifting the marginal cost
curve upward. This results in an optimal speed that is lower.)
    What if you can go to driving school and get a ticket removed? Will this cause you to drive faster, or
slower? (If the cost of going to driving school is lower than the cost of a ticket, it would lower the marginal
cost of driving faster.)
    Lastly, suppose you buy a vehicle that is very safe, perhaps a Volvo Sport Utility Vehicle. How will
that affect your driving behavior? (Since you feel safer, the lower risk of injury will lower the marginal
cost of speeding up, resulting in a faster optimal speed.)

                                                  References
Kloeden, C., McLean, A., Moore, V., and Ponte, G., 1997, “Travelling Speed and the Risk of Crash
Involvement,” NHMRC Road Accident Research Unit, The University of Adelaide, November 1997
http://raru.adelaide.edu.au/speed, accessed 08/31/2008.

				
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